spin waves in the ferromagnetic heavy rare ...jjensen/book/echap5.pdf186 5. spin waves in the...

5

SPIN WAVES IN THE FERROMAGNETICHEAVY RARE EARTHS

As discussed in Section 1.5, the exchange interaction dominates the mag-netic behaviour of the heavy rare earth metals, and the ordered momentsat low temperatures are consequently close to the saturation values. Theexcitations of such a system are spin waves, which may be viewed semi-classically as coupled precessions of the moments about their equilib-rium directions, with well-defined frequencies which are determined bythe phase relations between the precessing moments on different sites.From the viewpoint of quantum mechanics, these modes are magnons,which are linear combinations of single-ion excitations from the groundstate to the first excited molecular-field state, which is to a good approx-imation |Jz = J − 1>, with phase factors between the coefficients fordifferent ions which determine the dispersion relation Eq for the magnonenergy. A useful review of the excitations of magnetic systems has beengiven by Stirling and McEwen (1987).

These spin waves have been very extensively studied in the heavyrare earths, both experimentally and theoretically. In this chapter, weconsider the simplest case of the ferromagnet, in which all the sitesare equivalent. Since the magnetic heavy rare earths are all hcp, webegin by extending the earlier treatment of the linear response of theisotropic Heisenberg ferromagnet to this structure. These results areimmediately applicable to Gd, where the anisotropy is indeed negligible,with the consequence that the excitation spectrum is the simplest tobe found among the magnetic rare earths. Crystal-field and magneto-elastic anisotropies modify the excitation spectrum significantly, induc-ing an elliptical polarization of the precessing moments, and a spin-waveenergy gap at long wavelengths. To treat such systems, we employ linearspin-wave theory, determining the magnon energies via the Holstein–Primakoff transformation. We consider in particular the basal-planeferromagnet, comparing the calculated excitation spectrum throughoutwith experimental measurements on Tb, which has been very compre-hensively studied. The magnon energies and their temperature depen-dence are discussed, and the energy gap associated with the uniformspin-wave mode is treated in some detail and related to the macro-scopic magnetic anisotropy. The contribution to this energy gap of themagnetoelastic coupling, via the static deformation of the crystal, is then

182 5. SPIN WAVES IN THE FERROMAGNETIC RARE EARTHS

calculated and its dynamic manifestation in the magnon–phonon inter-action is discussed. Anisotropic two-ion coupling between the momentsalters the form of the dispersion relations, both quantitatively and, onoccasions, qualitatively. The classical dipole–dipole interaction, thoughweak, is highly anisotropic and long-ranged, and may therefore havean important influence at long wavelengths. Since its form is knownexactly, we can calculate its effects in detail, but we can say much lessabout the two-ion anisotropy in general. Its possible origins and symme-try are however discussed, and examples of observable effects to which itgives rise are presented. The mutual solubility of the rare earths allowsthe formation of an enormous variety of binary alloys, with magneticproperties which may be adjusted with the concentration. We show howthe excitation spectrum of such systems can be calculated by the virtualcrystal approximation and the coherent potential approximation, and il-lustrate the phenomena which may be observed by experiments on Tballoys. Finally, we consider the interaction between the conduction elec-trons and the localized 4f moments, and its influence on both the spinwaves and the conduction electrons themselves. The indirect-exchangeinteraction is derived more rigorously than in Section 1.4, and the life-time of the magnons due to electron scattering is deduced. The massenhancement of the conduction electrons is determined, and the effectsof magnetic ordering on the band structure, and of magnetic scatteringon the conductivity, are discussed.

5.1 The ferromagnetic hcp-crystal

In Chapter 3, we considered the linear response of a system of magneticmoments placed on a Bravais lattice and coupled by the Heisenberginteraction. We shall now generalize this treatment to the hexagonalclose-packed crystal structure of the heavy rare earth metals, in whichthere is a basis of two ions per unit cell, constituting two identical sub-lattices which, for convenience, we number 1 and 2. The surroundings ofthe atoms belonging to each of the two sublattices are identical, exceptfor an inversion. Introducing the following Fourier transforms:

Jss′ (q) =∑

j∈s′−subl.J (ij) e−iq·(Ri−Rj) ; i ∈ s-sublattice,

(5.1.1a)we have, for an hcp crystal,

J1(q) ≡ J11(q) = J22(q)J2(q) ≡ J12(q) = J21(−q) = J ∗21(q),

(5.1.1b)

where J1(q) is real. Defining the four Fourier transforms χss′(q, ω) ofthe susceptibility tensor equivalently to (5.1.1a), we obtain from the

5.1 THE FERROMAGNETIC HCP-CRYSTAL 183

RPA equation (3.5.7):

χ11(q, ω) = χo(ω)

{1 + J11(q)χ11(q, ω) + J12(q)χ21(q, ω)

}χ21(q, ω) = χ

o(ω){J21(q)χ11(q, ω) + J22(q)χ21(q, ω)

},

(5.1.2)

assuming that the MF susceptibility χ o(ω) is the same for all the sites,as in a paramagnet or a ferromagnet. These matrix equations may besolved straightforwardly, and using (5.1.1b) we find

χ11(q, ω) = D(q, ω)−1{1 − χ o(ω)J1(q)}χ

o(ω)

χ21(q, ω) = D(q, ω)−1 {χ o(ω)}2 J2(−q), (5.1.3a)

where

D(q, ω) ={1 − χ o(ω)J1(q)

}2 − {χ o(ω) |J2(q)|}2=(1 − χ o(ω) {J1(q) + |J2(q)|}

) (1 − χ o(ω) {J1(q) − |J2(q)|}

),

(5.1.3b)and, by symmetry,

χ22(q, ω) = χ11(q, ω) and χ12(q, ω) = χ21(−q, ω). (5.1.3c)If χ o(ω) contains only one pole, as in the case of the Heisenberg ferro-magnet, then D(q, ω)−1 in (5.1.3a) generates two poles, correspondingto the existence of both an acoustic and an optical mode at each q-vector.J2(0) must be real and, since it is also positive in a ferromagnet, theacoustic mode arises from the zero of the first factor in (5.1.3b), itsenergy therefore being determined by the effective coupling parameterJ1(q) + |J2(q)|. On the other hand, if J2(0) is negative, as it is inparamagnetic Pr, it is the second factor which gives the acoustic mode.The nomenclature results from the circumstance that the deviations ofthe moments from their equilibrium values are in phase in the acousticmode in the limit of q → 0, and it therefore dominates the neutroncross-section. The inelastic neutron scattering is determined by (4.2.2)and (4.2.3), i.e. by

χ(κ, ω) = 1N

∑ij

χ(ij, ω) e−iκ·(Ri−Rj) = 12

∑ss′

χss′ (κ, ω)

= D(κ, ω)−1{1 − χ o(ω)

(J1(κ) − 12 [J2(κ) + J2(−κ)]

)}χ

o(ω),(5.1.4)

where N = 2N0 is the number of atoms. Introducing κ = q + τ , withq lying in the primitive zone, we may write this result as a sum of theacoustic and optical response functions:

χAc(q, ω) ={1 − χ o(ω)(J1(q) + ν|J2(q)|)

}−1χ

o(ω)

χOp(q, ω) ={1 − χ o(ω)(J1(q) − ν|J2(q)|)

}−1χ

o(ω),(5.1.5)


where ν = ±1 denotes the sign of J2(0). J1(κ) = J1(q) is real, whereas

J2(κ) = J2(q) eiτ ·ρ = ν|J2(q)|eiϕ, (5.1.6a)

defining the phase ϕ = ϕ(κ), and ρ = d2 − d1 is the vector joining thetwo sublattices. In terms of these quantities, the susceptibility (5.1.4)may be written

χ(q + τ , ω) = 12 (1 + cosϕ)χAc(q, ω) +12 (1− cosϕ)χOp(q, ω). (5.1.6b)

The phase ϕ vanishes in the limit q → 0 if τ = 0, and the scatteringcross-section then only depends on the isolated pole in the acoustic re-sponse function, in accordance with our definition above. Introducingthe following lattice vectors of the hexagonal lattice:

a1 = (a, 0, 0) a2 =(−a

2,

√3a

2, 0)

a3 = (0, 0, c), (5.1.7a)

we find the corresponding reciprocal lattice vectors:

b1 =(2π

a,

2π√3a

,0)

b2 =(0, 4π√

3a, 0)

b3 =(0, 0, 2π

c

). (5.1.7b)

Since ρ =(a2

,a

2√

3,c

2

),

τ ·ρ = 4π3h+ 2π

3k+πl with τ = (hkl) = hb1 +kb2 + lb3. (5.1.8)

If q is parallel to the c-axis, J2(q) is real. The phase ϕ in (5.1.6) is thenτ ·ρ and, if the Miller indices h and k are both zero, ϕ = τ ·ρ = lπ. In thiscase, with κ in the c-direction, the inelastic scattering detects only theacoustic or the optical excitations, depending on whether l is respectivelyeven or odd, and no energy gap appears at the zone boundary, eventhough l changes, because J2(b3/2) = 0 by symmetry. We may thereforeuse a double-zone representation, in which the dispersion relation for theexcitations is considered as comprising a single branch extending twicethe distance to the Brillouin zone boundary, corresponding to an effectiveunit cell of height c/2. We shall generally use this representation whendiscussing excitations propagating in the c-direction.

5.1 THE FERROMAGNETIC HCP-CRYSTAL 185

Fig. 5.1. Spin-wave dispersion relations for Gd, after Koehler etal. (1970). The two atoms of the hcp structure give rise to acousticand optical branches. Since the single-ion anisotropy is negligible, the

acoustic mode rises quadratically from the origin.

Because L = 0, so that J = S, anisotropy effects are small in Gd,and it is therefore a good approximation to a Heisenberg ferromagnet.Using the above procedure to generalize (3.5.26) to the hcp structure,we obtain the two branches of the excitation spectrum

EAcq = 〈Jz〉{J1(0) + J2(0) − J1(q) − |J2(q)|

}EOpq = 〈Jz〉

{J1(0) + J2(0) − J1(q) + |J2(q)|

},

(5.1.9)

since J2(0) is positive. The dispersion relations measured by inelasticneutron scattering by Koehler et al. (1970) are shown in Fig. 5.1. Thisfigure illustrates the use of the double-zone representation when q isalong the c-axis, resulting in a single spin-wave branch. The renormal-ization predicted by the simple RPA theory, that Eq(T ) is proportionalto σ, is not followed very precisely. σ changes from about 0.97 at 78Kto 0.66 at 232K. As may be seen from Fig. 5.1, and from more exten-sive studies by Cable et al. (1985), the energies in the c-direction varyapproximately like σ0.5 at the largest wave-vectors, like σ in the mid-dle of the branch, and faster than σ at small wave-vectors. However, itis also evident from the figure that the form of J (q) changes with de-creasing magnetization, so some of the discrepancy between the simple


prediction and the observed behaviour at low temperatures may be dueto changes of J (q). At higher temperatures, the RPA renormalizationbreaks down completely. The spin-wave energy at the zone boundaryhas only fallen by about a factor two at 292K, very close to TC . Fur-thermore, strongly-broadened neutron peaks are observed even at 320K,well above the transition, close to the zone boundary in the basal plane,with energies of about kBTC . On the other hand, the low-energy spinwaves progressively broaden out into diffusive peaks as TC is approachedfrom below.

5.2 Spin waves in the anisotropic ferromagnet

In the heavy rare earth metals, the two-ion interactions are large andof long range. They induce magnetically-ordered states at relativelyhigh temperatures, and the ionic moments approach closely their sat-uration values at low temperatures. These circumstances allow us toadopt a somewhat different method, linear spin-wave theory, from thosediscussed previously in connection with the derivation of the correlationfunctions. We shall consider the specific case of a hexagonal close-packedcrystal ordered ferromagnetically, with the moments lying in the basalplane, corresponding to the low-temperature phases of both Tb and Dy.For simplicity, we shall initially treat only the anisotropic effects intro-duced by the single-ion crystal-field Hamiltonian so that, in the case ofhexagonal symmetry, we have

H =∑

i

[ ∑l=2,4,6

B0l Q0l (Ji) +B

66Q

66(Ji)− gµBJi ·H

]− 1

2

∑i�=j

J (ij)Ji ·Jj .

(5.2.1)The system is assumed to order ferromagnetically at low temperatures,a sufficient condition for which is that the maximum of J (q) occurs atq = 0. Qml (Ji) denotes the Stevens operator of the ith ion, but definedin terms of (Jξ, Jη, Jζ) instead of (Jx, Jy, Jz), where the (ξ, η, ζ)-axesare fixed to be along the symmetry a-, b- and c-directions, respectively,of the hexagonal lattice. The (x, y, z)-coordinate system is chosen suchthat the z-axis is along the magnetization axis, specified by the polarangles (θ, φ) in the (ξ, η, ζ)-coordinate system. Choosing the y-axis tolie in the basal plane, we obtain the following relations:

Jξ = Jz sin θ cosφ− Jx cos θ cosφ+ Jy sinφJη = Jz sin θ sinφ− Jx cos θ sinφ− Jy cosφJζ = Jz cos θ + Jx sin θ,

(5.2.2)

from which

Q02 = 3{J2z cos2 θ+J2x sin2 θ+(JzJx+JxJz) cos θ sin θ}−J(J+1). (5.2.3)

5.2 SPIN WAVES IN THE ANISOTROPIC FERROMAGNET 187

Initially we assume that 〈Jz〉 = J at T = 0, which implies that theground state is the product of |Jz = J >-states of the single ions. Inthis case, we find, consistently with eqn (2.2.14),

〈Q02〉 = = J (2)(3 cos2 θ − 1),

where, as before, J (n) = J(J − 12 ) · · · (J −n−1

2 ), and we have used theexpectation values 〈J 2z 〉 = J2, 〈J 2x 〉 = 12J and 〈JzJx〉 = 0. Analogously,though with considerably more labour, we can show that, for instance,

〈Q66〉 = = J (6) sin6 θ cos 6φ. (5.2.4)

For simplicity, we neglect for the moment B04 and B06 , and specifying

the direction of the magnetic field by the polar angles (θH , φH), we findthat the ground-state energy is, within this approximation,

U(T = 0) � N[B02J

(2)(3 cos2 θ − 1) +B66J (6) sin6 θ cos 6φ− gµBJH{cos θ cos θH + sin θ sin θH cos (φ− φH)} − 12J (0)J

2],

(5.2.5)where θ and φ are determined so that they minimize this expression. Inzero magnetic field, H = 0, (5.2.5) only gives two possibilities for θ, viz.θ = 0 for B02J

(2) < − 13 |B66 |J (6) or θ =π2 for B

02J

(2) > − 13 |B66 |J (6). Weshall here be concerned with the second case of θ = π2 , i.e. the basal-plane ferromagnet. In this case, the angle φ is determined by the signof B66 . The magnetic moments will be along an a- or a b-axis (φ = 0or φ = π2 ) if B

66 is respectively negative or positive. Having specified

the (approximate) ground state, we turn to the excitations, i.e. the spinwaves.

Instead of utilizing the standard-basis operators, defined by (3.5.11),we shall introduce a Bose operator ai for the ith ion, satisfying

[ai , a+j ] = δij ; [ai , aj ] = [a

+i , a

+j ] = 0, (5.2.6)

which acts on the |Jz >-state vector of this ion (the site index is sup-pressed) in the following way:

a |J > = 0 ; a |J −m> =√m |J −m+ 1> (5.2.7)

Holstein and Primakoff (1940) introduced the following representationof the angular momentum operators:

Jz = J − a+a

J+ =(2J − a+a

) 12 a

J− = a+(2J − a+a

) 12 .

(5.2.8)


If the usual basis vectors in the Hilbert space created by the Bose oper-ators are denoted by |n), i.e. a|n) =

√n|n− 1) where n = 0, 1, 2, · · · ,∞,

then by the definition (5.2.7), |n) = |Jz = J−n> for n = 0, 1, 2, · · · , 2J ,but there is no physical |Jz>-state corresponding to |n) when n > 2J .It is straightforward to see that the Bose representation (5.2.8) producesthe right matrix-elements of the angular momentum operators, as longas |n) is restricted to the physical part of the Hilbert space, n ≤ 2J ,but this representation presupposes the presence of an infinite numberof states. In the ferromagnetic case, the unphysical states are at highenergies, if J is large and T is low, and their influence on the thermalaverages is negligible. In this regime of J and T , the Holstein–Primakofftransformation is useful and the results derived from it are trustworthy.

In order to be able to treat the Bose operators under the squareroots in eqn (5.2.8), we shall utilize 1/J as an expansion parameter.This means that, instead of the J± given by (5.2.8), we shall use

J+ = (J−)† �√

2J(a− 1

4Ja+aa

). (5.2.9)

It is important here to realize that the expansion parameter is 1/J andnot, for instance, ‘the number of deviation operators’. If the latterwere the case, a well-ordered expansion of J+ (Lindg̊ard and Danielsen1974) would suggest instead J+ =

√2J{a−(1−

√1 − 1/2J)a+aa+ · · ·},

corresponding to a replacement of 14J in (5.2.9) by14J (1+

18J + · · ·). We

emphasize that we shall be expanding the reduced operators (1/J (l))Oml ,leaving no ambiguities either in (5.2.9) or in the following. Using eqn(5.2.9) and Jz = J − a+a, it is straightforward to express the Stevensoperators in terms of the Bose operators. For O02 , we get

O02 = 3J2z − J(J + 1) = 3(J − a+a)2 − J(J + 1)

= 2J(J − 12 ) − 6(J −12 ) a

+a+ 3a+a+aa

= 2J (2){1 − 3

Ja+a+ 3

2J2a+a+aa+ O(1/J3)

}.

(5.2.10)

Here we have used [a , a+] = 1 to arrange the operators in ‘well-ordered’products, with all the creation operators to the left, and in the last line1/J (2) has been replaced by 1/J2 in the term of second order in 1/J . Inthe same way, we obtain

O22 =12 (J

2+ + J

2−) = J

(2){ 1

J(a+a+ + aa)

+ 14J2

(a+a+ + aa− 2a+a+a+a− 2a+aaa) + O(1/J3)}.

(5.2.11)


The expression for Q02 is then determined using Q02(θ =π2 ) = −

12O

02 +

32O

22 . For the case of Q

66, we refer to Lindg̊ard and Danielsen (1974),

who have established the Bose operator expansion of the tensor operatorsup to the eighth rank. Introducing these expansions into (5.2.1), andgrouping the terms together according to their order in 1/J , we maywrite the Hamiltonian

H = H0 + H1 + H2 + · · · + H′, (5.2.12)

where H0 = U0 is the zero-order term, and

U0 = N[−B02J (2) +B66J (6) cos 6φ− gµBJH cos (φ− φH) − 12J

2J (0)],

(5.2.13)corresponding to (5.2.5), when we restrict ourselves to the case θ = θH =π/2. H1 comprises the terms of first order in 1/J , and is found to be

H1 =∑

i

[Aa+i ai +B

12 (a

+i a

+i + aiai)

]−∑ij

JJ (ij)(a+i aj − a+i ai),

(5.2.14)where the parameters A and B are

A = 1J

{3B02J

(2) − 21B66J (6) cos 6φ+ gµBJH cos (φ − φH)}

B = 1J

{3B02J

(2) + 15B66J(6) cos 6φ

}.

(5.2.15)

If we consider only the zero- and first-order part of the Hamiltonian,i.e. assume H � H0 + H1, it can be brought into diagonal form viatwo transformations. The first step is to introduce the spatial Fouriertransforms of J (ij), eqn (3.4.2), and of ai:

aq =1√N

∑i

ai e−iq·Ri ; a+q =

1√N

∑i

a+i eiq·Ri , (5.2.16)

for which the commutators are

[aq , a+q′ ] =

1N

∑i

e−i(q−q′)·Ri = δqq′.

In the case of an hcp lattice, with its two ions per unit cell, the situationis slightly more complex, as discussed in the previous section. However,this complication is inessential in the present context, and for simplicitywe consider a Bravais lattice in the rest of this section, so that the resultswhich we obtain are only strictly valid for excitations propagating in


the c-direction, for which the double-zone representation may be used.Introducing the Fourier transforms, we may write

H1 =∑q

[Aq a

+q aq +B

12 (a

+q a

+−q + aqa−q)

], (5.2.17)

withAq = A+ J{J (0) − J (q)}. (5.2.18)

H1 is quadratic in the Bose operators, and it can be diagonalized byperforming a Bogoliubov transformation. A new Bose operator αq isintroduced, such that

aq = uqαq − vqα+−q ; |uq|2 − |vq|2 = 1, (5.2.19)

in terms of which H0 + H1 is transformed into

H0 + H1 = U0 + U1 +∑q

Eq α+q αq, (5.2.20)

when uq and vq are adjusted appropriately. Here they can both bechosen to be real quantities, and are determined by the equation

(uq ± vq)2 = (Aq ±B)/Eq. (5.2.21)

The energy parameters are

U1 =1

2

∑q

(Eq −Aq) ; Eq =√A2q −B2. (5.2.22)

When B is different from zero, as occurs if either B02 or B66 is non-zero,

the product of the |Jiz = J >= |0)i-states is no longer the (MF) groundstate. Q02 and Q

66 give rise to couplings between the single-ion states

|J >, |J −2> etc. as reflected in the term proportional to B in (5.2.17).The new ground state established by the Bogoliubov transformationhas the energy U0 + U1 (= U0 −

∑qB

2/4Eq to leading order in B),which is always smaller than U0. The admixture of (predominantly) the|J − 2>-state into the ground state implies that the system is no longerfully polarized at T = 0, as assumed in (5.2.5). Using (5.2.19) and theconditions 〈αqαq〉 = 〈α+q α+q 〉 = 0, whereas

〈α+q αq〉 = nq =1

eβEq − 1 (5.2.23)


is the usual Bose population-factor, we find to first order in 1/J :

〈Jz〉 =〈J − 1

N

∑q

a+q aq〉

= J(1 −m), (5.2.24)

withm = 1

N

∑q

1

J〈a+q aq〉 =

1

N

∑q

mq

andmq =

1

J〈(uqα+q − vqα−q)(uqαq − vqα+−q)〉

= 1J

{u2qnq + v

2q(nq + 1)

}= 1

J

{AqEq

(nq +

12

)− 12},

(5.2.25)

which is positive and non-zero, even when nq = 0 at T = 0.The second-order contribution to the Hamiltonian is

H2 =∑

i

[B

1

8J(a+i a

+i + aiai) + C1a

+i a

+i aiai

+ C2(a+i a+i a

+i ai + a

+i aiaiai) + C3(a

+i a

+i a

+i a

+i + aiaiaiai)

]− 14

∑ij

J (ij)(2a+i a

+j aiaj − a+i a+j ajaj − a+i a+i aiaj

), (5.2.26)

withC1 = −

1

J2

(32B

02J

(2) − 105B66J (6) cos 6φ)

C2 = −1

J2

(34B

02J

(2) + 1954 B66J

(6) cos 6φ)

C3 =1

J2154 B

66J

(6) cos 6φ.

(5.2.27)

Introducing the Fourier transforms of the Bose operators in H2, we findstraightforwardly that

ih̄∂aq/∂t = [aq , H] � [aq , H1 + H2] = Aqaq +B(1 + 1

4J

)a+−q +

1

N

∑k,k′

[{−J (q − k′)+ 12J (k

′)+ 14J (k)+14J (q)+2C1

}a+k ak′aq+k−k′

+ C2{3a+k a

+−k′aq+k−k′ + a−kak′aq+k−k′

}+ 4C3 a+k a

+−k′a

+−q−k+k′

],

(5.2.28)for the operator [aq , H], which appears in the equation of motion of,for instance 〈〈aq ; a+q 〉〉. When the thermal averages of terms due to H2


are considered, the replacement of H by H0 + H1 in the density matrixonly gives rise to errors of higher-order in 1/J . Because H0 + H1 isquadratic in the Bose operators, this replacement results in a decouplingof the H2-terms (according to Wick’s theorem) which is equivalent to theRPA decoupling utilized previously. Hence, when considering thermalaverages, we have to leading order in 1/J , for instance,

a+k ak′aq+k−k′ � a+k 〈ak′aq+k−k′〉 + ak′〈a

+k aq+k−k′〉 + aq+k−k′〈akak′〉

= δk,−qa+−q〈ak′a−k′〉 + δk′,qaq〈a+k ak〉 + δk,k′aq〈a

+k ak〉,

(5.2.29)where the last line follows from the diagonality of H0 +H1 in reciprocalspace. We note that it is convenient here that the single-ion operators areexpressed as products of Bose operators which are well-ordered. Whenthis decoupling is introduced in (5.2.28), it reduces to

[aq , H] = Ãq(T ) aq + B̃q(T ) a+−q, (5.2.30)

where the effective, renormalized parameters are

Ãq(T ) = A+4JC1m+ 6JC2b+ J{J (0) − J (k)}(1 −m)

+ 1N

∑k

J{J (k) − J (k − q)}mk (5.2.31a)

and

B̃q(T ) =B(1 + 1

4J

)+ 2JC1b+6JC2m+12JC3b− 12J{J (0) −J (q)}b

+ 12N

∑k

J{J (0) − J (k)}bk +1

N

∑k

J{J (k) − J (k − q)}bk.

(5.2.31b)mk and bk are respectively the correlation functions (1/J)〈a+k ak〉 and(1/J)〈a+k a

+−k〉 = (1/J)〈aka−k〉, and m and b are the corresponding aver-

ages over k. Equation (5.2.30) implies that the operator [aq , H], in theequations of motion of any Green function involving aq, can be replacedby the expression on the right-hand side. The same result is obtained if,instead, H2 is neglected, and Aq and B in H1 are replaced by Ãq(T ) andB̃q(T ) in (5.2.17). Consequently, the system behaves as if the Hamilto-nian H0 +H1 +H2 is replaced by H̃0 + H̃1, which is similar to H0 +H1except for the introduction of the effective, temperature-dependent pa-rameters. The RPA decoupling (5.2.29) introduces errors in the Greenfunctions, but only in the third order of 1/J , and as it leads to an effec-tive Hamiltonian which is quadratic in the Bose operators, it is a validprocedure. This internal consistency of the theory to second order in


1/J means that the RPA contributions to the correlation functions arereliably estimated, and that all second-order contributions are includedwhen H̃0 +H̃1 is used, instead of H0 +H1, in the calculation of the ther-mal averages. We shall therefore use the following self-consistent expres-sions for the characteristic correlation functions, mk and bk, determinedstraightforwardly by utilizing the correspondence between H0 +H1 andH̃0 + H̃1:

mk =1J

{ Ãk(T )Ek(T )

(nk +

12

)− 12}, (5.2.32a)

corresponding to (5.2.25), and

bk = −1J

B̃k(T )Ek(T )

(nk +

12

). (5.2.32b)

In order to express the result in a convenient form, we rewrite one of thesecond-order terms in B̃q(T ) as

1

2N

∑k

J{J (0)−J (k)}bk = −12B(m+12J )−

12Ab+O(1/J

3), (5.2.33)

since, to leading order, J{J (0) − J (k)} = Ãk(T ) − A, and B̃k(T ) inbk can be approximated by B. We note that Aq and B are parametersof the order 1/J , as are m and b (at low temperatures). In additionto introducing (5.2.33) into (5.2.31b), it is adequate for calculating thespin-wave energies to define a transformed set of parameters:

Aq(T ) = Ãq(T ) + 12 B̃q(T ) b

Bq(T ) = B̃q(T ) + 12 Ãq(T ) b(5.2.34)

and these are then, to the order considered,

Aq(T ) = A+ 4JC1m+ 6JC2b + 12Bb

+J{J (0) − J (q)}(1 −m) + 1N

∑k

J{J (k) − J (k − q)}mk

(5.2.35a)and

Bq(T ) = B + 2JC1b+6JC2m+ 12JC3b− 12Bm

+ 1N

∑k

J{J (k) − J (k − q)}bk.(5.2.35b)

This transformation leaves the expression for the excitation energies un-changed, i.e.

Eq(T ) ={[Aq(T ) +Bq(T )][Aq(T ) −Bq(T )]

} 12 , (5.2.36)


when higher-order corrections are neglected. Inserting the eqns (5.2.15),(5.2.18), and (5.2.27) into (5.2.35), we finally obtain, at zero wave-vector,

A0(T ) −B0(T ) =1

J

{−36B66J (6)(1 − 20m+ 15b) cos6φ

+ gµBJH cos (φ− φH)}

(5.2.37a)

and

A0(T ) +B0(T ) =1

J

{6B02J

(2)(1 − 2m− b)

− 6B66J (6)(1 − 20m+ 5b) cos 6φ+ gµBJH cos (φ− φH)}, (5.2.37b)

and, at non-zero wave-vector,

Aq(T ) = A0(T )+J{J (0)−J (q)}(1−m)+1

N

∑k

J{J (k)−J (k−q)}mk

(5.2.38a)and

Bq(T ) = B0(T ) +1

N

∑k

J{J (k) − J (k − q)}bk. (5.2.38b)

The spin-wave energies deduced here, to second order in the expansionin 1/J , depend on temperature and on the crystal-field mixing of theJz-eigenstates, and both dependences are introduced via the two corre-lation functions mk and bk, given self-consistently by (5.2.32) in termsof the energy parameters. Bq(T ) vanishes if there is no anisotropy, i.e.if B02 and B

66 are zero. In the case of single-ion anisotropy, Bq(T ) is in-

dependent of q if the small second-order term in (5.2.38b) is neglected,nor does it depend on the magnetic field, except for the slight field-dependence which may occur via the correlation functions m and b.

When the spin-wave excitation energies have been calculated, it is astraightforward matter to obtain the corresponding response functions.Within the present approximation, the xx-component of the susceptibil-ity is

χxx(q, ω) = −1

4N

∑ij

〈〈(J+ + J−)i e−iq·Ri ; (J+ + J−)j eiq·Rj〉〉

= −J2

(1 − 12m−

14b)2〈〈aq + a+−q ; a+q + a−q〉〉.

(5.2.39)The Bogoliubov transformation, eqns (5.2.19) and (5.2.21), with theparameters replaced by renormalized values, then leads to

χxx(q, ω) = −J

2

(1 −m− 12 b

)Ãq(T ) − B̃q(T )Eq(T )

〈〈αq + α+−q ; α+q + α−q〉〉,


which is a simple combination of Bose Green-functions determined by(5.2.20), with Eq replaced by Eq(T ). Introducing these functions andthe parameters given by (5.2.34), we finally obtain

χxx(q, ω) = J(1 −m)Aq(T ) −Bq(T )E2q(T ) − (h̄ω)2

, (5.2.40a)

neglecting third-order terms. A rotation of the coordinate system byπ/2 around the z-axis changes the sign of Bq(T ), and hence we have

χyy(q, ω) = J(1 −m)Aq(T ) +Bq(T )E2q(T ) − (h̄ω)2

. (5.2.40b)

These results show that the ratio between the neutron-scattering inten-sities due to the spin-wave at q, neglecting Szz(q, ω), in the two caseswhere the scattering vector is perpendicular to the basal y–z plane andto the x–z plane is

Rq(T ) =Sxx(q, ω)Syy(q, ω)

∣∣∣∣h̄ω=±Eq(T )

=χxx(q, 0)χyy(q, 0)

=Aq(T ) −Bq(T )Aq(T ) +Bq(T )

.

(5.2.41)The measured intensities from Tb, which differ substantially from thosecalculated for the Heisenberg ferromagnet, agree well with this expres-sion, especially if the correction for anisotropic two-ion coupling is takeninto account (Jensen et al. 1975).

In the Heisenberg ferromagnet without rotational anisotropy, corre-sponding to Bq(T ) = 0, the elementary excitations at low temperaturesare circularly polarized spin waves, in which the local moments precessin circles around the equilibrium direction. In the presence of anisotropy,Rq(T ) differs from unity, and the excitations become elliptically polar-ized spin waves. The eccentricity of the ellipse depends on the wave-vector of the excited spin wave, and by definition Rq(T ) is the square ofthe ratio of the lengths of the principal axes which, at least to the orderin 1/J which we have considered, is equal to the ratio between the cor-responding static susceptibility components. So the static anisotropy isreflected, in a direct way, in the normal modes of the system. The result(5.2.41) justifies the transformation (5.2.34) by attributing observableeffects to the parameters Aq(T )±Bq(T ), whereas the parameters whichare defined via the Hamiltonian alone, here Ãq(T )± B̃q(T ), depend onthe particular Bose representation which is employed.

The longitudinal correlation function Szz(q, ω), which is neglectedabove, contains a diffusive mode at zero frequency, but no well-definednormal modes of non-zero frequency. There is inelastic scattering, but


the inelastic response, as well as the elastic mode, are purely of secondorder in 1/J and we shall not consider the longitudinal fluctuationsfurther here.

The method developed in this section may be utilized, essentiallyunchanged, to calculate the MF susceptibility χ o(ω) of the single sites.The result to first order in 1/J is:

χ oxx(ω) = 〈Jz〉A−B + hexE2ex − (h̄ω)2

χ oyy(ω) = 〈Jz〉A+B + hexE2ex − (h̄ω)2

χ oxy(ω) = −χ oyx(ω) = 〈Jz〉ih̄ω

E2ex − (h̄ω)2,

(5.2.42a)

where 〈Jz〉 is the MF expectation value of Jz , hex is the exchange field,and Eex is the energy of the first excited MF state:

hex = 〈Jz〉J (0) ; E2ex = (A+ hex)2 −B2. (5.2.42b)

Introducing this expression for χ o(ω) into the RPA equation (3.5.8), wemay derive χ(q, ω) by the same method as was used for the Heisenbergferromagnet in Section 3.5.2, in which case A = B = 0. The results forthe xx- and yy-components are then found to agree with eqn (5.2.40)to leading order in 1/J . To the next order in 1/J , the parameters arereplaced by renormalized values, but this procedure is not here easilygeneralized so as to become fully self-consistent. However, most of thecorrections may be included by substituting A0(T ) ± B0(T ) for A ± Bin the expression for χ o(ω), and the self-consistent value of 〈Jz〉 for itsMF value. The only terms which are not included in χ(q, ω) by thisprocedure, as we may see by a comparison with eqn (5.2.40), are theq-dependent contributions to Aq(T )±Bq(T ) determined by the k-sumsin (5.2.38). At low temperatures, these contributions are small andmay safely be neglected in systems with long-range interactions. Thisformulation therefore represents a valid alternative, which is useful forgeneralizing the linear spin-wave theory to the hcp structure, discussedin Section 5.1, or to the helically or conically ordered systems which wewill consider in Chapter 6.

As an example of the magnon dispersion relations for the anisotropicbasal-plane ferromagnet, we show in Fig. 5.2 experimental measurementson Tb at 4K (Mackintosh and Bjerrum Møller 1972). The principaldifferences between these results and the corresponding excitations forGd in Fig. 5.1 are the pronounced interactions which are observed be-tween the magnons and phonons, which we shall discuss in some detail in


Fig. 5.2. The spin-wave dispersion relations along the symmetry linesin the Brillouin zone for Tb. In contrast to Gd, the anisotropy gives rise toan energy gap at the origin, and there are large effects due to interactionswith the phonons. The third branch along, for example, ΓM may also bedue to phonon interactions, or it may be a manifestation of the breakingof the hexagonal symmetry by the ordered moment in a particular do-main, in the multi-domain sample.The lifting of the double degeneracyalong the line KH provides evidence for anisotropic two-ion coupling.


Section 5.4.2, and the appearance of an energy gap at long wavelengths.This gap has its origin in the magnetic anisotropy. Even thoughthe exchange energy required to excite a magnon vanishes in the long-wavelength limit, work is still required to turn the moments away fromthe easy direction against the anisotropy forces. If we neglect the smallterms due to the sums over k in (5.2.38), the dispersion relation alongthe c-axis in zero field becomes, from eqns (5.2.36–38),

Eq(T ) ={[A0(T ) +B0(T ) + 〈Jz〉{J (0) − J (q)}]

× [A0(T ) −B0(T ) + 〈Jz〉{J (0) − J (q)}]} 1

2 .(5.2.43)

For an arbitrary direction in the zone, this relation is generalized anal-ogously to eqn (5.1.9), giving rise again to acoustic and optical modes.From the dispersion relations, the magnon density of states and J (q)may readily be determined and hence, by a Fourier transform, the nom-inal Heisenberg exchange interaction J (ij) between moments on differ-ent atomic sites (Houmann 1968). The energy gap at zero wave-vectoris given by

E0(T ) ={[A0(T ) +B0(T )][A0(T ) −B0(T )]

} 12 , (5.2.44)

and as we shall see in the next section, it is proportional to the geo-metrical mean of the axial- and hexagonal-anisotropy energies. We shallreturn to the dependence of this energy gap on the temperature and themagnetoelastic effects in the following two sections.

5.3 The uniform mode and spin-wave theory

The spin-wave mode at zero wave-vector is of particular interest. Incomparison with the Heisenberg ferromagnet, the non-zero energy ofthis mode is the most distinct feature in the excitation spectrum of theanisotropic ferromagnet. In addition, the magnitude of the energy gapat q = 0 is closely related to the bulk magnetic properties, which maybe measured by conventional techniques. We shall first explore the con-nection between the static magnetic susceptibility and the energy of theuniform mode, leading to an expression for the temperature dependenceof the energy gap. In the light of this discussion, we will then considerthe general question of the validity of the spin-wave theory which wehave presented in this chapter.

5.3.1 The magnetic susceptibility and the energy gapThe static-susceptibility components of the bulk crystal may be deter-mined as the second derivatives of the free energy

F = U − TS = − 1β

lnZ. (5.3.1)

5.3 THE UNIFORM MODE AND SPIN-WAVE THEORY 199

The specific heat C may be derived in a simple way, within our currentspin-wave approximation, by noting that the excitation spectrum is thesame as that for a non-interacting Bose system, so that the entropyis fully determined by the statistics of independent bosons of energiesEq(T ):

S = kB∑q

[(1 + nq) ln (1 + nq) − nq lnnq

], (5.3.2)

and hence

C = T∂S/∂T = kBT∑q

(dnq/dT ) ln {(1 + nq)/nq},

or, with nq =[eβEq(T ) − 1

]−1,C =

∑q

Eq(T ) dnq/dT

= β∑q

nq(1 + nq)Eq(T ){Eq(T )/T − ∂Eq(T )/∂T

},

(5.3.3)

as in (3.4.17).The first derivative of F with respect to the angles θ and φ can be

obtained in two ways. The first is to introduce S, as given by (5.3.2)into (5.3.1), so that

∂F

∂θ=∂U

∂θ−∑q

Eq(T )∂nq∂θ

=∂U

∂θ

∣∣∣∣mq,bq

+∑q

(∂U

∂mq

∂mq∂θ

+∂U

∂bq

∂bq∂θ

− Eq(T )∂nq∂θ

)

=∂U

∂θ

∣∣∣∣mq,bq

, (5.3.4)

as it can be shown that ∂U/∂mq = JÃq(T ) and ∂U/∂bq = JB̃q(T ),when U = 〈H0 +H1 +H2〉, and hence that each term in the sum over qin the second line of (5.3.4) vanishes, when (5.2.32) is used. This resultis only valid to second order in 1/J . However, a result of general validityis

∂F/∂θ =〈∂H/∂θ

〉, (5.3.5)

as discussed in Section 2.1, in connection with eqn (2.1.5). The two dif-ferent expressions for ∂F/∂θ, and corresponding expressions for ∂F/∂φ,agree if H in (5.3.5) is approximated by H0 + H1 + H2, i.e. to secondorder in 1/J . However, the results obtained up to now are based on the


additional assumption, which we have not stated explicitly, that H′ inthe starting Hamiltonian (5.2.12) is negligible. H′ is the sum of the termsproportional to Stevens operators Oml with m odd, and it includes for in-stance the term 3B02(JzJx+JxJz) cos θ sin θ associated with B02Q02 in eqn(5.2.3). H′ vanishes by symmetry if the magnetization is along a high-symmetry direction, i.e. θ = 0 or π/2 and φ is a multiple of π/6. In thesecases, the results obtained previously are valid. If the magnetization isnot along a high-symmetry direction, H′ must be taken into account.The first-order contributions arise from terms proportional to (1/J)1/2

in H′, which can be expressed effectively as a linear combination of Jxand Jy. In this order, 〈∂H′/∂θ〉 = 0 therefore, because 〈Jx〉 = 〈Jy〉 = 0by definition. For a harmonic oscillator, corresponding in this systemto the first order in 1/J , the condition for the elimination of terms inthe Hamiltonian linear in a and a+ coincides with the equilibrium con-dition ∂F/∂θ = ∂F/∂φ = 0. Although the linear terms due to H′ canbe removed from the Hamiltonian by a suitable transformation, termscubic in the Bose operators remain. Second-order perturbation theoryshows that, if H′ is non-zero, 〈∂H′/∂θ〉 and the excitation energies in-clude contributions of the order 1/J2. Although it is straightforward tosee that H′ makes contributions of the order 1/J2, it is not trivial tocalculate them. The effects of H′ have not been discussed in this con-text in the literature, but we refer to the recent papers of Rastelli et al.(1985, 1986), in which they analyse the equivalent problem in the caseof a helically ordered system.

In order to prevent H′ from influencing the 1/J2-contributions de-rived above, we may restrict our discussion to cases where the mag-netization is along high-symmetry directions. This does not, however,guarantee that H′ is unimportant in, for instance, the second deriva-tives of F . In fact ∂〈∂H′/∂θ〉/∂θ ∝ O(1/J2) may also be non-zero whenθ = 0 or π/2, and using (5.3.4) we may write

Fθθ =∂2F

∂θ2=∂2U

∂θ2

∣∣∣∣mq,bq

+ O(1/J2)

=〈∂2∂θ2

(H0 + H1 + H2)〉

+ O(1/J2) ; θ = 0, π2,

(5.3.6a)

and similarly

Fφφ =〈∂2∂φ2

(H0 + H1 + H2)〉

+ O(1/J2) ; φ = p π6, (5.3.6b)

where the corrections of order 1/J2 are exclusively due to H′. Here wehave utilized the condition that the first derivatives of mq and bq vanishwhen the magnetization is along a symmetry direction.


The derivatives Fθθ and Fφφ are directly related to the static sus-ceptibilities, as shown in Section 2.2.2. When θ0 =

π2 , we obtain from

eqn (2.2.18)

χxx(0, 0) = N〈Jz〉2/Fθθ ; χyy(0, 0) = N〈Jz〉2/Fφφ. (5.3.7)

These results are of general validity, but we shall proceed one step furtherand use F (θ, φ) for estimating the frequency dependence of the bulksusceptibilities. When considering the uniform behaviour of the system,we may to a good approximation assume that the equations of motionfor all the different moments are the same:

h̄∂〈J〉/∂t = 〈J〉 × h(eff). (5.3.8)

By equating it to the average field, we may determine the effective fieldfrom

F = F (0) −N〈J〉 · h(eff), (5.3.9a)

corresponding to N isolated moments placed in the field h(eff). The freeenergy is

F = F (θ0, φ0) +12Fθθ(δθ)

2 + 12Fφφ(δφ)2 −N〈J〉 · h, (5.3.9b)

and, to leading order, δθ = −〈Jx〉/〈Jz〉 and δφ = −〈Jy〉/〈Jz〉. Hence

hx(eff) = −1N

∂F

∂〈Jx〉= hx −

1NFθθ

〈Jx〉〈Jz〉2

, (5.3.10a)

and similarly

hy(eff) = hy −1NFφφ

〈Jy〉〈Jz〉2

. (5.3.10b)

Introducing a harmonic field applied perpendicular to the z-axis intoeqn (5.3.8), we have

ih̄ω〈Jx〉 =1

N〈Jz〉Fφφ〈Jy〉 − hy〈Jz〉

ih̄ω〈Jy〉 = −1

N〈Jz〉Fθθ〈Jx〉 − hx〈Jz〉,

(5.3.11)

and ∂〈Jz〉/∂t = 0, to leading order in h. Solving the two equations forhx = 0, we find

χyy(0, ω) = 〈Jy〉/hy =1N

FθθE20(T ) − (h̄ω)2

, (5.3.12a)


and, when hy = 0,

χxx(0, ω) =1N

FφφE20(T ) − (h̄ω)2

, (5.3.12b)

where the uniform-mode energy is

E0(T ) =1

N〈Jz〉{FθθFφφ

}1/2. (5.3.13)

This result for the uniform mode in an anisotropic ferromagnet wasderived by Smit and Beljers (1955). It may be generalized to an arbitrarymagnetization direction by defining (θ, φ) to be in a coordinate systemin which the polar axis is perpendicular to the z-axis (as is the case here),and by replacing FθθFφφ by FθθFφφ − F 2θφ if Fθφ �= 0.

The introduction of the averaged effective-field in (5.3.8) corre-sponds to the procedure adopted in the RPA, and a comparison of theresults (5.3.12–13) with the RPA result (5.2.40), at q = 0 and ω = 0,shows that the relations

A0(T ) −B0(T ) =1

N〈Jz〉Fφφ

A0(T ) +B0(T ) =1

N〈Jz〉Fθθ

(5.3.14)

must be valid to second order in 1/J . In this approximation, A0(T ) ±B0(T ) are directly determined by that part of the time-averaged two-dimensional potential, experienced by the single moments, which isquadratic in the components of the moments perpendicular to the mag-netization axis. The excitation energy of the uniform mode is thus pro-portional to the geometric mean of the two force constants characterizingthe parabolic part of this potential. Since A0(T )±B0(T ) are parametersof order 1/J , the second-order contributions of H′ in (5.3.6), which arenot known, appear only in order 1/J3 in (5.3.14), when the magnetiz-ation is along a high-symmetry direction.

B02 does not appear in A0(T ) − B0(T ), and this is in accordancewith eqn (5.3.14), as Q02 is independent of φ. Considering instead theθ-dependence, we find that the contribution to Fθθ is determined by〈∂2Q02

∂θ2〉

=〈− 6(J2z − J2x) cos 2θ − 6(JzJx + JxJz) sin 2θ

〉θ=π/2

= 3〈O02 −O22〉. (5.3.15)

From (5.2.10) and (5.2.11), the thermal average is found to be

〈O02 −O22〉 = 2J (2)〈1 − 3

Ja+a+ 3

2J2a+a+aa

− 12J

(1 + 14J )(aa+ a+a+) + 1

4J2(a+aaa+ a+a+a+a)

〉,


or

〈O02 −O22〉 = 2J (2){1− 3m+ 3m2 + 32b2 − (1 + 14J )b+

32mb+O(1/J

3)}.(5.3.16)

Hence, according to (5.3.6a) and (5.3.14), the B02 -term contributes tothe spin-wave parameter A0(T ) +B0(T ) by

3B02〈O02 −O22〉/〈Jz〉 � 6B02J (2)(1 − 3m− b)/J(1 −m)� 6B02J (2)(1 − 2m− b)/J,

in agreement with (5.2.37b). When b is zero, this result is consistent withthe classical Zener power-law (Zener 1954), 〈Oml 〉 ∝ δm0 σl(l+1)/2, whereσ = 1 −m is the relative magnetization, since, to the order considered,〈O02 − O22〉b=0 = 〈O02〉b=0 = 2J (2)(1 − m)3. If we include the diagonalcontribution of third order in m or 1/J to 〈O02〉 in (5.3.16), the resultdiffers from the Zener power-law, but agrees, at low temperatures, withthe more accurate theory of Callen and Callen (1960, 1965) discussed inSection 2.2. The results of the linear spin-wave theory obtained abovecan be utilized for generalizing the theory of Callen and Callen to thecase of an anisotropic ferromagnet. The elliptical polarization of the spinwaves introduces corrections to the thermal expectation values, whichwe express in the form

〈O02 −O22〉 = 2J (2)Î5/2[σ] η−1+ , (5.3.17)

where the factor Îl+1/2[σ] represents the result (2.2.5) of Callen andCallen, and where η± differs from 1 if b is non-zero. The two correlationfunctions m and b are determined through eqn (5.2.32), in terms of theintermediate parameters Ãk(T ) ± B̃k(T ), but it is more appropriate toconsider instead

mo =1NJ

∑k

{Ak(T )Ek(T )

(nk +

12

)− 12}

bo = −1NJ

∑k

Bk(T )Ek(T )

(nk +

12

),

(5.3.18)

defined in terms of the more fundamental parameters. The transforma-tion (5.2.34) then leads to the following relations:

mo + 12J = m+12J −

12b

2 and bo = b− 12b(m+12J ).

Separating the two contributions in (5.3.16), we find

b̃ ≡ 〈O22〉/〈O02〉 � (1 + 14J )b(1 −m)−3/2, (5.3.19a)


which, to the order calculated, may be written

b̃ =(1 − 1

2J

)−1 boσ2, (5.3.19b)

whereσ = 〈Jz〉/J = 1 −m = 1 −mo − 12 bob̃. (5.3.20)

The function η± is then determined in terms of b̃ as

η± = (1 ± b̃)(1 − 12 b̃2). (5.3.21)

The spin-wave theory determines the correlation functions σ and η±to second order in 1/J , but for later convenience we have includedsome higher-order terms in (5.3.20) and (5.3.21). It may be straightfor-wardly verified that the thermal expectation values of 〈O02 − O22〉 givenby (5.3.16) and (5.3.17) agree with each other to order 1/J2. In the ab-sence of anisotropy, the latter has a wider temperature range of validitythan the former, extending beyond the regime where the excitations canbe considered to be bosons. This should still be true in the presence ofanisotropy, as long as b̃ is small.

The combination of the spin-wave theory and the theory of Callenand Callen has thus led to an improved determination of the thermalaverages of single-ion Stevens operators, as shown in Figs. 2.2 and 2.3.The quantity O02 − O22 was chosen as an example, but the procedure isthe same for any other single-ion average. It is tempting also to utilizethis improvement in the calculation of the excitation energies, and therelation (5.3.14) between the free energy and the spin-wave parametersA0(T ) ± B0(T ) is useful for this purpose. Neglecting the modificationsdue to H′ in (5.3.6), i.e. using Fθθ � 〈∂2H/∂θ2〉 and similarly for Fφφ,we find from (5.3.14) the following results:

A0(T )−B0(T ) = −1

Jσ36B66J

(6)Î13/2[σ]η−15− cos 6φ+gµBH cos (φ − φH)

(5.3.22a)and

A0(T )+B0(T ) =1

Jσ

[6B02J

(2)Î5/2[σ]η−1+ − 60B04J (4)Î9/2[σ]η7−η−1+

+ 210B06J(6)Î13/2[σ]η

18− η

−1+ − 6B66J (6)Î13/2[σ]η−30− η−25+ cos 6φ

]+ gµBH cos (φ− φH), (5.3.22b)

which for completeness include all contributions from the starting Hamil-tonian (5.2.1). The spin-wave spectrum at non-zero wave-vectors isadjusted accordingly by inserting A0(T ) ± B0(T ) given above, instead


of (5.2.37), in eqns (5.2.36), (5.2.38), and (5.2.40). If the out-of-planeanisotropy is stronger than the in-plane anisotropy, as in Tb and Dy, Bis positive and b̃ is negative. This means that η+ and η− are respectivelysmaller and greater than 1 (for small b̃ ), with the result that the axialcontributions to A0(T ) +B0(T ) are increased, whereas the planar con-tribution to A0(T ) − B0(T ) is diminished, due to b̃. This is consistentwith the fact that the out-of-plane fluctuations are suppressed in com-parison with the in-plane fluctuations by the anisotropy. Hence we find,as a general result, that the elliptical polarization of the spin waves en-hances, in a self-consistent fashion, the effects of the anisotropy. We notethat Q66, which depends on both θ and φ, contributes to both anisotropyparameters, but that the anisotropy of the fluctuations affects the twocontributions differently.

If b̃ and the k-sums in (5.2.38) are neglected, the above result forthe spin-wave energies Eq(T ) reduces to that derived by Cooper (1968b).The modifications due to the non-spherical precession of the moments,b̃ �= 0, were considered first by Brooks et al. (1968) and Brooks (1970),followed by the more systematic and comprehensive analysis of Brooksand Egami (1973). They utilized directly the equations of motion ofthe angular-momentum operators, without introducing a Bose repre-sentation. Their results are consistent with those above, except thatthey did not include all the second-order modifications considered here.We also refer to Tsuru (1986), who has more recently obtained a re-sult corresponding to eqn (5.2.31), when B66 is neglected, using a varia-tional approach. The procedure outlined above essentially follows thatof Lindg̊ard and Danielsen (1974, 1975), which was further developedby Jensen (1975). This account only differs from that given by Jensenin the use of η± instead of b̃ as the basis for the ‘power-law’ general-ization (and by the alternative choice of sign for B and b̃) and, moreimportantly, by the explicit use of 1/J as the expansion parameter.

As illustrated in Fig. 5.1 for Gd, and in Fig. 5.3 for Tb, the observedtemperature dependence of the spin-wave spectrum is indeed substan-tial, both in the isotropic and the anisotropic ferromagnet. In the caseof Tb, the variation of the exchange contribution is augmented by thetemperature dependence of the anisotropy terms, which is reflected pre-dominantly in the rapid variation of the energy gap at q = 0. A com-parison of Figs. 5.1 and 5.3 shows that the change in the form of J (q)appears to be more pronounced in Tb than in Gd. In Tb, the variationof J (q) with q at a particular temperature is also modified if the mag-netization vector is rotated from the b-axis to a hard a-axis (Jensen etal. 1975). Most of these changes with magnetization can be explainedas the result of two-ion anisotropy, which we will consider in Section 5.5.Anisotropic two-ion terms may also affect the energy gap. In addition,


Fig. 5.3. The temperature dependence of the dispersion relations forthe unperturbed spin waves in the c-direction in Tb. Both the energy gapand the q-dependence renormalize with temperature. The results havebeen corrected for the magnon–phonon interaction, and the lines show the

calculated energies.

the magnetoelastic coupling introduces qualitatively new effects, not de-scribable by eqn (5.3.22), to which we will return after a short digressionto summarize our understanding of the spin-wave theory.

5.3.2 The validity of the spin-wave theory

In presenting the spin-wave theory, we have neglected phenomena whichfirst appear in the third order of 1/J , most importantly the finite life-times of the excitations. In the presence of anisotropy, when B is dif-ferent from zero, the total moment is not a conserved quantity, since[∑

i Jiz , H ] �= 0, unlike in the Heisenberg model. On the microscopic


plane, this means that the number of spin-wave excitations, i.e. magnons,is not necessarily conserved in a scattering process. In contrast to the be-haviour of the isotropic ferromagnet, the linewidths do not therefore van-ish at zero temperature, although energy conservation, combined withthe presence of an energy gap in the magnon spectrum, strongly limitthe importance of the allowed decay processes at low temperatures.

The two-ion interactions are assumed to involve only tensor op-erators of the lowest rank, so that these terms in the 1/J-expansiononly have small numerical factors multiplying the Bose operator prod-ucts. Therefore, if J is large, as in heavy rare earth-ions, the third-orderterms due to the exchange coupling, which are neglected in the spin-wave theory, are expected to be small, as long as the number of excitedmagnons is not very large. The weak influence, at low temperatures,of the higher-order contributions of the exchange coupling is also indi-cated by a comparison with the low-temperature expansion of Dyson(1956) of the free energy in a Heisenberg ferromagnet with only nearest-neighbour interactions, also discussed by Rastelli and Lindg̊ard (1979).If A = B = 0, the results derived earlier, to second order in 1/J , areconsistent with those of Dyson, except that we have only included theleading-order contribution, in the Born approximation or in powers of1/J , to the T 4-term in the magnetization and in the specific heat. Thehigher-order corrections to the T 4-term are significant if J = 12 , but ifJ = 6 as in Tb, for example, they only amount to a few per cent of thisterm and can be neglected.

If only the two-ion terms are considered, the RPA decoupling ofthe Bose operator products (5.2.29) is a good approximation at largeJ and at low temperatures. However, this decoupling also involves anapproximation to the single-ion terms, and these introduce qualitativelynew features into the spin-wave theory in the third order of 1/J . Forexample, the C3-term in (5.2.26) directly couples the |Jz = J > stateand |J − 4>, leading to an extra modification of the ground state notdescribable in terms of B or η±. Furthermore, the Bogoliubov trans-formation causes the (Jx, Jy)-matrix elements between the ground stateand the third excited state to become non-zero. This coupling thenleads to the appearance of a new pole in the transverse susceptibilities,in addition to the spin-wave pole, at an energy which, to leading order,is roughly independent of q and close to that of the third excited MFlevel, i.e. 3Eqo(T ), with qo defined as a wave-vector at which J (qo) = 0.A qualitative analysis indicates that the third-order contribution to e.g.χxx(0, 0), due to this pole, must cancel the second-order contribution ofH′ to Fθθ in the relation (5.3.12b) between the two quantities. Hencethe approximation Fθθ � 〈∂2H/∂θ2〉, used in (5.3.22), corresponds tothe neglect of this additional pole.


The higher-order exchange contributions can thus be neglected atlow temperatures, if J is large. This condition is not, however, sufficientto guarantee that the additional MF pole is unimportant, and the spin-wave result (5.3.22), combined with (5.2.36), (5.2.38), and (5.2.40), canonly be trusted as long as the modification of the ground state, due tothe single-ion anisotropy, is weak. This condition is equivalent to therequirement that |̃b| be much less than 1. The regime within which thespin-wave theory is valid can be examined more closely by a comparisonwith the MF-RPA theory. In the latter, only the two-ion interactionsare treated approximately, whereas the MF Hamiltonian is diagonalizedexactly. The MF-RPA decoupling utilized in Section 3.5 leads here toa cancellation of the k-sums in (5.3.38), and to a replacement of thecorrelation functions mo and bo by their MF values

mo � mMFo =1J

{Aqo(T )Eqo(T )

(nqo +

12

)− 12}, (5.3.23)

with a similar expression for bMFo . The wave-vector qo is defined asabove, such that J (qo) = 0. If the single-ion anisotropy is of second rankonly, including possibly a Q22-term as well as the Q

02-term of our specific

model, all the predictions obtained with the MF-RPA version of the spin-wave theory agree extremely well with the numerical results obtainedby diagonalizing the MF Hamiltonian exactly, even for relatively largevalues of |bMFo | (≈ 0.1). Even though 1/J is the expansion parameter,the replacement of (1 + 12J ) by (1−

12J )

−1 in (5.3.19b) extends the goodagreement to the limit J = 1, in which case the MF Hamiltonian can bediagonalized analytically.

The applicability of the 1/J-expansion for the anisotropy is muchmore restricted if terms of high rank, such as Q66, dominate. This is asimple consequence of the relatively greater importance of the contribu-tions of higher-order in 1/J , like for instance the C3-term in (5.2.26),for higher-rank anisotropy terms. We have analysed numerically mod-els corresponding to the low-temperature phases of Tb and Er, whichinclude various combinations of anisotropy terms with ranks between 2and 6. In the case of the basal-plane ferromagnet Tb, we find that the1/J-expansion leads to an accurate description of the crystal-field effectson both the ground-state properties and the excitation energies. TheMF-RPA excitation-energies calculated with the procedure of Section3.5 differ relatively only by ∼ 10−3 at T = 0 from those of the spin-wavetheory (Jensen 1976c). We furthermore find that this good agreementextends to non-zero temperatures, and that the 1/J-expansion is still ac-ceptably accurate when σ � 0.8. Consequently, the effective power-lawspredicted by the spin-wave theory at low temperatures (Jensen 1975)are valid.


The renormalization of the anisotropy parameters appearing in thespin-wave energies, in the second order of 1/J , is expected to be some-what more important in the conical phase of Er than in Tb. In Er,the moments are not along a symmetry direction (they make an angleof about 28◦ with the c-axis) and the second-order modifications dueto H′ in (5.2.12) might be expected to be important. The 1/J-resultsdo not allow a precise estimate of the second-order contributions, butby introducing two scaling parameters, one multiplying the exchangeterms by σ, and the other scaling the constant crystal-field contributionin the 1/J-expression for the spin-wave energies in the cone phase, itis possible (Jensen 1976c) to give an accurate account of the excitationenergies derived by diagonalizing the MF Hamiltonian exactly, the rela-tive differences being only of the order 10−2. The two scaling parametersare found to have the expected magnitudes, although σ turns out to beslightly smaller (� 0.94 in the model considered) than the relative mag-netization predicted by the MF Hamiltonian (σMF � 0.98). An analysisof the MF Hamiltonian shows that the excitations can be described interms of an elliptical precession of the single moments, as expected, butsurprisingly the ellipsoid lies in a plane with its normal making an angle(� 33◦) with the c-axis which differs from the equilibrium cone-angle(� 28◦), so the polarization of the spin waves is not purely transverse.In terms of the 1/J-expansion, this modification of the excited statescan only be produced by H′. This observation indicates that H′ has sig-nificant effects in Er, since it explains the difference between σ and σMF,as σ becomes equal to σMF if the angle appearing in the renormalizedspin-wave energies is considered to be that defining the excited states,i.e. 33◦, rather than the equilibrium value.

We may conclude that the 1/J-expansion is a valid procedure fordescribing the low-temperature magnetic properties of the heavy rareearth metals. This is an important conclusion for several reasons. Tofirst order in 1/J , the theory is simple and transparent. It is thereforefeasible to include various kinds of complication in the model calcula-tions and to isolate their consequences. This simplicity is retained inthe second order of 1/J , as long as H′ can be neglected, in which casethe first-order parameters are just renormalized. Accurate calculationsof the amount of renormalization of the different terms may be quiteinvolved, but because of the long range of the two-ion interactions inthe rare earth metals, the MF values of mo and bo utilized above nor-mally provide good estimates. The spin-wave theory in the harmonicapproximation, to first order in 1/J , has been employed quite exten-sively in the literature, both for analysing experimental results and invarious theoretical developments. It is therefore fortunate that theseanalyses are not invalidated, but only modified, or renormalized, by the


presence of moderate anisotropy. However, it is necessary to be awarethat the renormalization itself may cause special effects not expectedin the harmonic approximation, as the amount of renormalization maychange when the system is perturbed by an external magnetic field orpressure, or when the temperature is altered.

There have been attempts (Lindg̊ard 1978, and references therein)to construct an analytical spin-wave theory starting with a diagonaliza-tion of the MF Hamiltonian. In principle, this should be an appropriatestarting-point, since the ground state is closer to the MF ground-statethan to the fully polarized state, as soon as the planar anisotropy be-comes significant. As in the model calculations discussed above, the MFHamiltonian can be diagonalized numerically without difficulty, but inthis form the method is non-analytical and the results are not easilyinterpretable. In order to diagonalize the MF Hamiltonian analytically,one is forced to make a perturbative expansion, unless J is small. Ifthe MF Hamiltonian is expressed in the |Jz >-basis, the natural ex-pansion parameter is ∼ |Bqo/Aqo | � 2J |bo| at T = 0. The use of thisexpansion parameter and the 1/J-expansion considered above lead toidentical results in the limit 2J |bo| � 1 (Rastelli and Lindg̊ard 1979).However, the expansion parameter is not small when the anisotropy ismoderately large (2J |bo| � 0.35 in Tb at T = 0), which severely limitsthe usefulness of this procedure as applied by Lindg̊ard (1978, 1988)to the analysis of the spin waves in the anisotropic heavy rare earths.It gives rise to a strong renormalization of all the leading-order spin-wave-energy parameters, which are thus quite sensitive, for example,to an external magnetic field, and it is extremely difficult to obtain areasonable estimate of the degree of renormalization. In contrast, the1/J-expansion leads, at low temperatures, to results in which only thehigh-rank terms (which are quite generally of smaller magnitude thanthe low-rank terms) are renormalized appreciably, and the amount ofrenormalization can be determined with fair accuracy. In the numericalexample corresponding to Tb, the B66 -term is renormalized by −38% atT = 0, according to the spin-wave theory, which agrees with the valueobtained by diagonalizing the MF Hamiltonian exactly, as indicated inFig. 2.3.

To recapitulate, we have developed a self-consistent RPA theory forthe elementary excitations in a ferromagnet, i.e. the spin waves, validwhen the magnetization is close to its saturation value. The major com-plication is the occurrence of anisotropic single-ion interactions, whichwere treated by performing a systematic expansion in 1/J . To firstorder in 1/J , the theory is transparent and simple, and it is straightfor-wardly generalized to different physical situations. Much of the simplic-ity is retained in second order, as long as the magnetization is along a

5.4 MAGNETOELASTIC EFFECTS 211

symmetry axis, but the first-order parameters are replaced by effectivevalues. These effective parameters are determined self-consistently interms of the spin-wave parameters Aq(T )±Bq(T ), which depend on T ,and on an eventual applied magnetic field. One advantage of the use of1/J as the expansion parameter is that the second-order modificationsare smallest for the low-rank couplings, which are quite generally alsothe largest terms. If the magnetization is not along a symmetry axis,the elementary excitations may no longer be purely transverse. Thisadditional second-order phenomenon may, however, be very difficult todetect experimentally within the regime of validity of the second-orderspin-wave theory.

5.4 Magnetoelastic effects

The magnetoelastic coupling between the magnetic moments and thelattice modifies the spin waves in two different ways. The static de-formations of the crystal, induced by the ordered moments, introducenew anisotropy terms in the spin-wave Hamiltonian. The dynamic time-dependent modulations of the magnetic moments furthermore interferewith the lattice vibrations. We shall start with a discussion of thestatic effects, and then consider the magnon–phonon interaction. Themagnetoelastic crystal-field Hamiltonian was introduced in Section 1.4,where the different contributions were classified according to the symme-try of the strain parameters. The two-ion coupling may also change withthe strain, as exemplified by eqn (2.2.32). We shall continue consider-ing the basal-plane ferromagnet and, in order to simplify the discussion,we shall only treat the low-rank magnetoelastic couplings of single-ionorigin. In the ferromagnetic case, the magnetoelastic two-ion couplingsdo not introduce any effects which differ qualitatively from those dueto the crystal-field interactions. Consequently, those interactions whichare not included in the following discussion only influence the detaileddependence of the effective coupling parameters on the magnetizationand, in the case of the dynamics, on the wave-vector.

5.4.1 Magnetoelastic effects on the energy gapThe static effects of the α-strains on the spin-wave energies may beincluded in a straightforward manner, by replacing the crystal-field pa-rameters in (5.2.1) with effective strain-dependent values, i.e. B02 →B02 + B

(2)α1 �α1 + B

(2)α2 �α2, with α-strains proportional to 〈Q02〉. Equiva-

lent contributions appear in the magnetic anisotropy, discussed in Sec-tion 2.2.2. This simplification is not possible with the γ- or the �-straincontributions, because these, in contrast to the α-strains, change thesymmetry of the lattice. When θ = π/2, the �-strains vanish, and the


γ-strain part of the magnetoelastic Hamiltonian is given by eqn (2.2.23):

Hγ =∑

i

[12cγ(�

2γ1 + �

2γ2) −Bγ2{Q22(Ji)�γ1 +Q−22 (Ji)�γ2}

−Bγ4{Q44(Ji)�γ1 −Q−44 (Ji)�γ2}].

(5.4.1)

The equilibrium condition, ∂F/∂�γ = 0, leads to eqn (2.2.25) for thestatic strains �γ . The static-strain variables are distinguished by a barfrom the dynamical contributions �γ − �γ . The expectation values ofthe Stevens operators may be calculated by the use of the RPA theorydeveloped in the preceding section, and with θ = π/2 we obtain, forinstance,

〈Q22〉 = 〈12 (O02 +O

22) cos 2φ+ 2O

−12 sin 2φ〉 = J (2)Î5/2[σ]η−1− cos 2φ

〈Q−22 〉 = 〈12 (O02 +O

22) sin 2φ− 2O−12 cos 2φ〉 = J (2)Î5/2[σ]η−1− sin 2φ.

(5.4.2)We note that 〈O−12 〉 vanishes only as long as H′ in (5.2.12) can be ne-glected. Introducing the magnetostriction parameters C and A via eqn(2.2.26a), when θ = π/2,

�γ1 = C cos 2φ− 12A cos 4φ�γ2 = C sin 2φ+ 12A sin 4φ,

(5.4.3)

and calculating 〈Q±44 〉, we obtain

C = 1cγBγ2J

(2)Î5/2[σ]η−1−

A = − 2cγBγ4J

(4)Î9/2[σ]η−6− ,

(5.4.4)

instead of eqn (2.2.26b), including the effects of the elliptical preces-sion of the moments. The equilibrium conditions allow us to split themagnetoelastic Hamiltonian into two parts:

Hγ(sta) =∑

i

[12cγ(�

2γ1 + �

2γ2) −Bγ2{Q22(Ji)�γ1 +Q−22 (Ji)�γ2}

−Bγ4{Q44(Ji)�γ1 −Q−44 (Ji)�γ2}], (5.4.5)

depending only on the static strains, and

Hγ(dyn) =∑

i

[12cγ{(�γ1 − �γ1)

2 + (�γ2 − �γ2)2}

−(Bγ2{Q22(Ji) − 〈Q22〉} +Bγ4{Q44(Ji) − 〈Q44〉}

)(�γ1 − �γ1)

−(Bγ2{Q−22 (Ji) − 〈Q−22 〉} −Bγ4{Q−44 (Ji) − 〈Q−44 〉}

)(�γ2 − �γ2)

](5.4.6)


depending only on the dynamical part of the strains.To leading order, the magnetoelastic energy is determined by the

static part (5.4.5), corresponding to eqn (2.2.27). Hγ influences theequilibrium condition determining φ and, in the spin-wave approxima-tion (H′ is neglected), we have

1N

∂F

∂φ=

1N

〈∂∂φ

{H + Hγ}〉� 1N

〈∂∂φ

{H + Hγ(sta)}〉

= − 6B66J (6)Î13/2[σ]η−15− sin 6φ+ gµBHJσ sin (φ − φH)

+2cγC(�γ1 sin 2φ− �γ2 cos 2φ) − 2cγA(�γ1 sin 4φ+ �γ2 cos 4φ),(5.4.7)

or, using the equilibrium values of �γ1 and �γ2,

1N

∂F

∂φ= gµBJσ

{H sin (φ− φH) − 16H̃c sin 6φ

}, (5.4.8a)

with the definition

gµBH̃c = 36κ66/(Jσ) = 36

{B66J

(6)Î13/2[σ]η−15− +

12cγCA

}/(Jσ).

(5.4.8b)If H = 0, the equilibrium condition ∂F/∂φ = 0 determines the sta-ble direction of magnetization to be along either a b-axis or an a-axis,depending on whether H̃c is positive or negative respectively.

The additional anisotropy terms introduced by Hγ and proportionalto the static strains, as for instance the term −Bγ2Q22(Ji)�γ1 in (5.4.5),contribute to the spin-wave energies. Proceeding as in Section 5.3, wefind the additional contributions to A0(T ) ±B0(T ) in (5.3.22), propor-tional to the static γ-strains,

∆{A0(T ) +B0(T )}=

cγJσ

{2C2 +A2η−8+ η

−4− − CA(2 + η−8+ η−4− ) cos 6φ

}η−1+ η−

∆{A0(T ) −B0(T )} =cγJσ

{4C2 + 4A2 − 10CA cos 6φ

}. (5.4.9)

The contribution to A0(T ) − B0(T ) is expressible directly in terms ofthe strain-parameters, C and A, without the further correction factorsnecessary for A0(T )+B0(T ). By using H̃c and the non-negative quantity

Λγ =4cγJσ

(C2 +A2 + 2CA cos 6φ), (5.4.10)

we can write the total spin-wave parameter

A0(T ) −B0(T ) = Λγ − gµBH̃c cos 6φ+ gµBH cos (φ − φH). (5.4.11)


This parameter does not obey the relation (5.3.14) with the secondderivative Fφφ of the free energy. A differentiation ∂F/∂φ, as givenby (5.4.8), with respect to φ shows that (5.3.14) accounts for the lasttwo terms in (5.4.11), but not for Λγ . A calculation from (5.4.7) of thesecond derivative of F , when the strains are kept constant, instead ofunder the constant (zero) stress-condition assumed above, yields

A0(T ) −B0(T ) =1

NJσ

∂2F

∂φ2

∣∣∣∣�=�

= Λγ +1

NJσFφφ, (5.4.12)

which replaces (5.3.14). The relation (5.3.14), determining A0(T ) −B0(T ), was based on a calculation of the frequency dependence of thebulk susceptibility and, as we shall see later, it is the influence of thelattice which invalidates this argument. The Λγ term was originally sug-gested by Turov and Shavrov (1965), who called it the ‘frozen lattice’contribution because the dynamic strain-contributions were not consid-ered. However, as we shall show in the next section, the magnon–phononcoupling does not change this result.

The modifications caused by the magnetoelastic γ-strain couplingsare strongly accentuated at a second-order phase transition, at whichFφφ vanishes. Let us consider the case where H̃c is positive, H̃c =|H̃c| ≡ Hc, i.e. the b-axis is the easy axis. If a field is applied along ana-axis, φH = 0, then the magnetization is pulled towards this direction,as described by eqn (5.4.8):

H = Hcsin 6φ6 sinφ

= Hc(1 − 163 sin

2 φ+ 163 sin4 φ)cosφ, (5.4.13)

as long as the field is smaller than Hc. At the critical field H = Hc,the moments are pulled into the hard direction, so that φ = 0 and thesecond derivative of the free energy,

Fφφ = NgµB{H cosφ−Hc cos 6φ}Jσ, (5.4.14)

vanishes. So a second-order phase transition occurs at H = Hc, and theorder parameter can be considered to be the component of the momentsperpendicular to the a-axis, which is zero for H ≥ Hc. An equally goodchoice for the order parameter is the strain �γ2, and these two possibili-ties reflect the nature of the linearly coupled magnetic–structural phasetransition. The free energy does not contain terms which are cubic inthe order parameters, but the transition might be changed into one offirst-order by terms proportional to cos 12φ, e.g. if σ or η±, and therebyH̃c, depend sufficiently strongly on φ (Jensen 1975). At the transition,eqn (5.4.11) leads to

A0(T ) −B0(T ) = Λγ at H = Hc, (5.4.15)


which shows the importance of the constant-strain contribution Λγ . Itensures that the spin-wave energy gap E0(T ), instead of going to zeroas |H − Hc|1/2, remains non-zero, as illustrated in Fig. 5.4, when thetransition at H = Hc is approached. Such a field just cancels the macro-scopic hexagonal anisotropy, but energy is still required in the spin waveto precess the moments against the strain field of the lattice.

By symmetry, the γ-strains do not contain terms linear in (θ − π2 ),and the choice between constant-stress and constant-strain conditionstherefore has no influence on their contribution to the second derivativeof F with respect to θ, at θ = π/2. Consequently, the γ-strains donot change the relation between A0(T ) + B0(T ) and Fθθ, given by eqn(5.3.14). The ε-strains vanish at θ = π/2, but they enter linearly with(θ − π2 ). Therefore they have no effect on A0(T ) + B0(T ), but theycontribute to Fθθ. To see this, we consider the ε-strain part of theHamiltonian, eqn (2.2.29):

Hε =∑

i

[12cε(�

2ε1 + �

2ε2) −Bε1{Q12(Ji)�ε1 +Q−12 (Ji)�ε2}

]. (5.4.16)

The equilibrium condition is

�ε1 =1

cεBε1〈Q12〉 = 14Hε sin 2θ cosφ,

in terms of the magnetostriction parameter Hε. In the basal-plane fer-romagnet, �ε1 and �ε2 both vanish. The transformation (5.2.2) leadsto

Q12 =14 (O

02 −O22) sin 2θ cosφ−O12 cos 2θ cosφ+O−12 cos θ sinφ

+ 12O−22 sin θ sinφ, (5.4.17)

and Q−12 is given by the same expression, if φ is replaced by φ− π2 . Thisimplies that

Hε =4

cεBε1〈14 (O

02 −O22)〉 =

2

cεBε1J

(2)Î5/2[σ]η−1+ . (5.4.18)

The static ε-strains are zero and do not contribute to the spin-waveparameters A0(T ) ± B0(T ), but they affect the second derivative of F ,with respect to θ, under zero-stress conditions and, corresponding to(5.4.12), we have

A0(T ) +B0(T ) =1

NJσ

∂2F

∂θ2

∣∣∣∣�=�

= Λε +1

NJσFθθ, (5.4.19)


withΛε =

cε4Jσ

H2ε , (5.4.20)

where Λε in (5.4.19) just cancels the ε-contribution to Fθθ/(NJσ) de-termined from eqn (2.2.34).

The dependence of the magnon energy gap in Tb on magnetic fieldand temperature has been studied in great detail by Houmann et al.(1975a). They expressed the axial- and hexagonal-anisotropy energiesof eqn (5.2.44) in the form

A0(T ) ±B0(T ) = P0(±) − P6(±) cos 6φ+ gµBH cos (φ− φH) (5.4.21)

and, by a least-squares fitting of their results, some of which are shownin Fig. 5.4, they were able to deduce the values of the four parametersP0,6(±), shown as a function of magnetization in Fig. 5.5. According toeqns (5.3.22) and (5.4.9), these parameters are given at low temperaturesby:

P0(+) ={6B02J

(2) − 60B04J (4) + 210B06J (6) + cγ(2C2 +A2)}/J (a)

P6(+) ={6B66J

(6) + 3cγCA}/J (b)

P0(−) = 4cγ{C2 +A2

}/J (c)

P6(−) ={36B66J

(6) + 10cγCA}/J, (d)

(5.4.22)where, for convenience, we have set the renormalization parameters σand η± to unity. These expressions for the parameters P0,6(±) are de-rived from a particular model. In general, additional contributions mayappear due to other magnetoelastic interactions, and to anisotropic two-ion couplings. Nevertheless, within the RPA, the relations between thespin-wave energy parameters A0(T ) ± B0(T ) and the bulk anisotropyparameters, (5.4.12) and (5.4.19) combined with (5.3.7), should still bevalid. The values of the anisotropy parameters, and their temperaturedependences, determine the static magnetic and magnetoelastic proper-ties, and can thus be obtained from bulk measurements on single crys-tals. A comparison between such static parameters and the dynamic val-ues P0,6(±), derived from the field dependence of the spin-wave energygap, can therefore elucidate the extent to which the spin-wave theory ofthe anisotropic ferromagnet is complete and correct.

Such a comparison has been made by Houmann et al. (1975a). Theaxial-anisotropy parameter P0(+)+P6(+), when the moments are alongthe easy axis, agrees with the values deduced from torque and mag-netization experiments, to within the rather large uncertainties of the


Fig. 5.4. The dependence of the square of the magnon energy gapin Tb on the internal magnetic field. Open symbols represent resultsfor the field in the hard direction, and closed symbols are for the easydirection. The non-zero value of the gap at the critical field, which justturns the moments into the hard direction, is due to the constant-straincontribution Λγ . The full lines are least-squares fits of the theoretical ex-pressions for the energy gap, given in the text, to the experimental results.

latter. The basal-plane anisotropies, as determined from the criticalfield Hc and the magnetoelastic γ-strain parameters, are well establishedby bulk measurements. Here P0(−) agrees, within the small combineduncertainties, with that derived from (5.4.22c) and (5.4.11), both inmagnitude and temperature dependence. On the other hand, the smallparameter P6(−) differs from the static value, so that

δ6(−) ≡ P6(−) − gµBH̃c + 8cγCA/(Jσ) (5.4.23a)

is found to be non-zero. A part of this discrepancy may be explained bya twelve-fold anisotropy term, but this would also affect P0(−), and isexpected to decrease more rapidly with increasing temperature than theexperiments indicate. Within the accuracy of the experimental results,the non-zero value of δ6(−) is the only indication of an additional renor-malization of the spin-wave energy gap, compared with that derivedfrom the second derivatives of the free energy.


Fig. 5.5. Anisotropy parameters in Tb as a function of the relativemagnetization, deduced from results of the type illustrated in Fig. 5.4.

The σ3-dependence of P0(+) on temperature is consistent with theσ2-renormalization of the dominant two-fold term in (5.4.22a) predictedby the Callen–Callen theory, but a comparison with the studies of diluteTb-alloys by Høg and Touborg (1975) suggests that a large part of theaxial anisotropy may have its origin in the two-ion coupling. The effectof the two-ion anisotropy is directly apparent in that part of the axialanisotropy P6(+) which depends on the orientation of the moments inthe basal plane. If only single-ion anisotropy of the type which we haveconsidered is important, P6(+) in (5.4.22b) is directly related to the crit-ical field necessary to turn the moments into the hard direction. How-ever, the experimental value of P6(+) bears little relation to gµBH̃c/6,even having the opposite sign. We can express this discrepancy by theparameter ∆M , defined by

∆M = P6(+) − gµBH̃c/6. (5.4.23b)

The influence of ∆M can be directly seen in the results of Fig. 5.4, sinceit is responsible for the difference between the slopes when the field isapplied in the easy and hard directions. Although it could in principle bedue to higher-rank γ-strain magnetoelastic terms, the large magnitude


of ∆M , compared to the contributions of C and A to the energy gap,effectively precludes this possibly. We must therefore ascribe it to two-ion anisotropy.

In the analysis of the field dependence of the magnon energy gap,the possible dependences of the renormalization parameters σ and η±on magnetic field and the orientation of the moments were neglectedat zero temperature, but included at non-zero temperatures, assumingthe different parameters effectively to be functions of σ only. In thecase of Dy, the zero-temperature change of the renormalization as afunction of φ is of some importance (Egami 1972; Jensen 1975; Egamiand Flanders 1976), whereas in Tb we have estimated by various meansthat both approximations are justified. There are some indications thatthere might be a systematic error involved in the determination of theφ-dependent energy-gap parameters P6(±), possibly arising from theinfluence of the classical dipole forces on the inelastic neutron-scatteringat long wavelength, discussed in Section 5.5.1. An extrapolation of theresults found at non-zero wave-vectors to q = 0 suggests that bothP6(+) and P6(−) may be about a factor of two smaller than shownin Fig. 5.5. If this were the case, ∆M would still be too large to beexplained by the γ-strain couplings, but δ6(−) would be reduced almostto the level of the experimental uncertainties. Otherwise a non-zerovalue of δ6(−) can only be explained by theories beyond the RPA, e.g.by effects, proportional to the frequency, due to the interaction betweenthe spin-waves and the electron-hole pair-excitations of the conductionelectrons.

5.4.2 The magnon–phonon interactionThe displacement of the ith ion from its equilibrium position, δRi =u(Ri), can be expanded in normal phonon coordinates in the usual way:

u(Ri) =∑νk

Fνk(βνk + β+ν−k)e

ik·Ri , (5.4.24a)

with

F νk,α =[

h̄

2NMωνk

] 12

fνk,α. (5.4.24b)

M is the mass of the ions and fνk,α is the α-component of the phonon-polarization vector. βνk is the phonon-annihilation operator and ωνkthe corresponding phonon frequency, where ν denotes one of the three(acoustic) branches. The polarization vectors are normalized and aremutually orthogonal: ∑

α

(fνk,α)∗fν

′k,α = δνν′ . (5.4.24c)


For simplicity, we assume that there is only one ion per unit cell, butthe results we shall derive are also applicable to the hcp lattice, at leastfor the acoustic modes at long wavelengths. In this limit Hγ(dyn), eqn(5.4.6), augmented by the kinetic energy of the ions, is adequate fordiscussing dynamical effects due to the γ-strains, if �αβ are replaced bytheir local values

�αβ(i) = �αβ +i

2

∑νk

(kαFνk,β + kβF

νk,α)(βνk + β

+ν−k)e

ik·Ri . (5.4.25)

We shall initially concentrate on the most important dynamical effects,and consider only the inhomogeneous-strain terms involving Stevens op-erators with odd m. Assuming for the moment that φ = pπ2 , we obtainthe contribution −Bγ2{−2O−12 (Ji) cos 2φ}(�γ2(i)−�γ2) from eqn (5.4.6),and a corresponding term in Bγ4. Introducing the spin-deviation oper-ators through (5.2.8) and (5.2.9), we obtain, to leading order in m andb,

Bγ2O−12 (Ji) = J

(2)Bγ2i√2J

{a+i − ai −

5

4J(a+i a

+i ai − a+i aiai)

}= J (2)Bγ2

i√2J

(1 − 52m+

54 b)(a+i − ai)

= cγCi√2J

(1 + 12m+

14 b)(a+i − ai)

= icγC∑q

[Aq(T ) +Bq(T )

2NJσEq(T )

] 12

(α+q − α−q) e−iq·Ri ,

(5.4.26)utilizing the RPA decoupling (5.2.29) and introducing the (renormal-ized) magnon operators α+q and α−q, analogously with (5.2.39) and(5.2.40). The Bγ4-term is treated in the same way, and introducingthe phonon-operator expansion of the strains (5.4.25) into (5.4.6), wefind that H + Hγ leads to the following Hamiltonian for the system ofmagnons and phonons:

Hmp =∑k

Ek(T )α+kαk+

∑νk

{h̄ωνkβ

+νkβνk+W

νk (α

+k −α−k)(βνk+β

+ν−k)

}(5.4.27)

with a magnon–phonon interaction given by

W νk = −cγ√N(k1F

νk,2+k2F

νk,1)[Ak(T ) + Bk(T )

2JσEk(T )

] 12

(C cos 2φ+A cos 4φ).

(5.4.28)This Hamiltonian includes the part of Hγ which is linear in the magnonoperators when φ = pπ2 . The effects of the static deformations are


included in Ek(T ) through (5.4.11). In general, W νk couples all threephonon modes with the magnons. A simplification occurs when k isalong the 1- or 2-axis, i.e. when k is either parallel or perpendicular tothe magnetization vector. In this case, W νk is only different from zerowhen ν

spin waves in the ferromagnetic heavy rare ...jjensen/book/echap5.pdf186 5. spin waves in the...

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